Higher-genus mirror symmetry for Calabi–Yau threefolds

Let $(X,X^\vee)$ be a mirror pair of Calabi–Yau threefolds, and choose a large complex structure limit on the complex moduli space $\mathcal{M}$ of $X$. Let $\mathcal{L}$ be the vacuum line bundle, let $g\geq 1$, and let $\phi^0(z)$ and the mirror map be the period and coordinates determined at the chosen large complex structure limit. Higher-genus mirror symmetry. There exists a $C^\infty$ section $\mathcal{F}_g(z,\bar z)\in\Gamma_{C^\infty}(\mathcal{M},\mathcal{L}^{2-2g})$, called the genus-$g$ topological string amplitude, satisfying the recursive BCOV holomorphic anomaly equations

$$\partial_i\partial_{\bar j}\mathcal{F}_1=\frac{1}{2}C_{ijk}C^{kl}_{\bar j}+\left(1-\frac{\chi(X^\vee)}{24}\right)G_{i\bar j},$$ $$\partial_{\bar i}\mathcal{F}_g=\frac{1}{2}C^{jk}_{\bar i}\left(D_jD_k\mathcal{F}_{g-1}+\sum_{r=1}^{g-1}D_j\mathcal{F}_r\,D_k\mathcal{F}_{g-r}\right)\qquad(g\geq2).$$

Moreover, a holomorphic-limit procedure produces a holomorphic section $F_g(z)\in\Gamma(\mathcal{M},\mathcal{L}^{2-2g})$, and the Gromov–Witten potential $\tt F_g(t)$ of $X^\vee$ satisfies

$$\tt F_g(t)=(\phi^0(z))^{2g-2}F_g(z)$$

under the mirror map. This formulation is a higher-genus extension of genus-zero mirror symmetry, but the existence and mathematical construction of the amplitudes, the anomaly equations, and the holomorphic-limit identification remain open in general; the source notes that rigorous results are available only in limited settings.